Methods: The proton PBA consisted of a central axis term and an off axis term. The central axis term was determined from a central axis depth dose profile of a Monte Carlo simulated proton beam in water and was scaled by a mass stopping power ratio to account for other materials. The off axis term was determined from Fermi-Eyges scattering theory with material-dependent scattering powers to calculate the lateral spread of the proton beam in heterogeneous media. The nuclear halo dose, which was caused by large angle and non-elastic scattering events, was modeled using two terms: a Gaussian distribution and a Cauchy-Lorentz distribution. Depth-dependent widths and amplitudes of each distribution were determined by fitting a simulated 1-mm x 1-mm pencil beam in water. The PBA was evaluated in approximately 30 test phantoms containing bone and/or air heterogeneities at 4 energies and for 2 field sizes. Agreement between PBA and Monte Carlo simulations of the test conditions was quantified by computing the percentage of points within 2 percent dose difference or 1 mm distance to agreement.

Results: With the improved nuclear halo model, PBA calculations showed better than of 97% of dose points within 2% or 1 mm of MC distributions for all geometries examined. For phantoms containing laterally infinite heterogeneities, agreement between PBA and MC distributions was 100% at 2% or 1mm. For phantoms containing laterally finite heterogeneities, agreement was at least 97%. The points failing were due to the central axis approximation of the PBA in regions not influenced by the nuclear halo model.

Conclusions: The nuclear halo model developed in this work improves the agreement of the PBA with MC simulations in heterogeneous phantoms, particularly in low-dose regions that can be important for scanned-beam proton therapy.